Question

The input and output of an LTI system are x (t) = e^-3t u (t) and y (t) = e^-t u (t). The differential equation which characterizes the system is ___________

(a) \( \frac{dy(t)}{dt} + y(t) = \frac{dx(t)}{dt} + 3x(t)\)

(b) \( \frac{dy(t)}{dt} + 2y(t) = \frac{dx(t)}{dt} + 3x(t)\)

(c) \( \frac{dy(t)}{dt} – y(t) = \frac{dx(t)}{dt} + 3x(t)\)

(d) \( \frac{dy(t)}{dt} – 2y(t) = \frac{dx(t)}{dt} + 3x(t)\)

I have been asked this question by my college director while I was bunking the class.

My query is from Exponential Fourier Series and Fourier Transforms topic in chapter Fourier Series of Signals and Systems

Answer 1

Correct choice is (a) \( \frac{dy(t)}{dt} + y(t) = \frac{dx(t)}{dt} + 3x(t)\)

Easiest explanation: X (s) = \( \frac{1}{s+3}\), Y (s) = \( \frac{1}{s+1}\)

∴ H(s) = \(\frac{Y(s)}{X(s)} = \frac{1/(s+1)}{1/(s+3)} = \frac{s+3}{s+1}\)

Now, s Y(s) + Y(s) = s X(s) + 3 X(s)

So, the differential equation together with the condition of initial rest that characterizes the system is \( \frac{dy(t)}{dt} + y(t) = \frac{dx(t)}{dt} + 3x(t)\).